Aereo Inc.'s upstart TV streaming service has provoked a legal onslaught from broadcast networks. But even if it wins that fight, it still has to overcome more-pedestrian issues, like making sure it can pay for the electricity it needs.

The service depends on tiny antennas assigned to each of its individual users, who rent them to stream broadcast TV channels over the Web. Each of those antennas, which Aereo warehouses in centralized facilities, uses five to six watts of power. On their own, that isn't a whole lot—a...